AIMS Mathematics, 4(2): 242–253. DOI:10.3934/math.2019.2.242 Received: 08 December 2018 Accepted: 14 February 2019 http://www.aimspress.com/journal/Math Published: 14 March 2019 Research article A new approach to generate all Pythagorean triples Anthony Overmars1, Lorenzo Ntogramatzidis2 and Sitalakshmi Venkatraman1;∗ 1 Department of Information Technology, Melbourne Polytechnic, VIC, Australia 2 Department of Mathematics and Statistics, Curtin University, Perth (WA), Australia * Correspondence: Email:
[email protected]; Tel: +61892663143. Abstract: This paper revisits the topic of Pythagorean triples with a different perspective. While several methods have been explored to generate Pythagorean triples, none of them is complete in terms of generating all the triples without repetitions. Indeed, many existing methods concentrate on generating primitive triples but do not cater to non-primitives. By contrast, the approach presented in this paper to parameterise the Pythagorean triples generates all of the triples in a unique way, i.e., without repetitions. We also explore the relation of this new parameterisation with the Pythagorean family of odd triples and with the Platonic family of even triples. Keywords: Pythagorean triples; Diophantine equations; Euclidean triples; Platonic triples; right angle triangles Mathematics Subject Classification: 11A41, 11A51 1. Introduction The Pythogorean theorem states as follows: For any right-angled triangle, the square of the hypotenuse c equals the sum of squares of/on the two (shorter) legs lengths a and b, which is written as a2 + b2 = c2 and is among the Diophantine equations [1]. Several different proofs have been given for the Pythagorean theorem, see e.g., [2–4]. It was also shown that the converse of the theorem to be true.